Optimal. Leaf size=229 \[ \frac{\sqrt{2} (A b-a B) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{b d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{b d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.222851, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2756, 2665, 139, 138} \[ \frac{\sqrt{2} (A b-a B) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{b d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right )}{b d \sqrt{\cos (c+d x)+1} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2756
Rule 2665
Rule 139
Rule 138
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx &=\frac{B \int (a+b \cos (c+d x))^{5/3} \, dx}{b}+\frac{(A b-a B) \int (a+b \cos (c+d x))^{2/3} \, dx}{b}\\ &=-\frac{(B \sin (c+d x)) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{b d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)}}-\frac{((A b-a B) \sin (c+d x)) \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{b d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)}}\\ &=\frac{\left ((-a-b) B (a+b \cos (c+d x))^{2/3} \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{5/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{b d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)} \left (-\frac{a+b \cos (c+d x)}{-a-b}\right )^{2/3}}-\frac{\left ((A b-a B) (a+b \cos (c+d x))^{2/3} \sin (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{2/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{b d \sqrt{1-\cos (c+d x)} \sqrt{1+\cos (c+d x)} \left (-\frac{a+b \cos (c+d x)}{-a-b}\right )^{2/3}}\\ &=\frac{\sqrt{2} (a+b) B F_1\left (\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)}{b d \sqrt{1+\cos (c+d x)} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}+\frac{\sqrt{2} (A b-a B) F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^{2/3} \sin (c+d x)}{b d \sqrt{1+\cos (c+d x)} \left (\frac{a+b \cos (c+d x)}{a+b}\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 1.87, size = 259, normalized size = 1.13 \[ \frac{3 (a+b \cos (c+d x))^{2/3} \left (5 B \left (a^2-b^2\right ) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right )-(2 a B+5 A b) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right )+5 b^2 B \sin (c+d x)\right )}{25 b^2 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.221, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\cos \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\cos \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]